Feynman Path Integral Formulation

116 4 Hamiltonian and Wheeler-DeWitt Equationand Φ[g ij ] some new wave functional to be determined. From the original Wheeler-DeWitt equation one then obtains a new set of equations for the wavefunctionalΦ[g ij ],{−2iG ij,klδSδg ijδδ 2 S− iG ij,klδg kl δg ij δg kl−16πG · G ij,klδ 2δg ij δg kl+ Ĥ φ }Φ[g ij (x)] = 0 . (4.77){}δ2ig ij ∇ k + Ĥ φ iΦ[g ij (x)] = 0 . (4.78)δg jkThe next step consists in approximating (again, in analogy with the semiclassicalexpansion in non-relativistic quantum mechanics), the solution by neglectingsecond derivative terms δ 2 S/δg 2 and δ 2 /δg 2 terms in the equations for Φ[g ij ](Wheeler, 1964). The latter step is usually justified by regarding (or assuming) theback-reaction of quantum matter on the gravitational field as small.The resulting truncated Wheeler-deWitt equations then become, to first order inδ/δg,{−2iG ij,klδSδg ij{2ig ij ∇ k}δ+ Ĥ φ Φ[g ij (x)] = 0δg kl}δδg jk+ Ĥ φ iΦ[g ij (x)] = 0 .(4.79)Furthermore the wavefunction Φ[g ij (x)] is now evaluated along a solution of theclassical field equations g ij (x,t). This means that S[g ij ] is first determined from asolution of the classical Hamilton-Jacobi equationswith [see Eq. (4.73)]∂ t g ij = NG ij,klδSδg kl− 2(D i N j + D j N i ) , (4.80)D i ≡− 2 i ∇ jδδg ij, (4.81)after which the relevant derivatives δS/δg are inserted in Eq. (4.77).To make further progress, one need to be more specific about the form of thelapse (N) and shift (N i ) functions. One can show that Φ[g ij ] satisfies an evolutionequation of the type∫∂∂t Φ(t) = d 3 δx ∂ t g ij (x)δg ij (x) Φ[g mn] , (4.82)